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  • 1
    Electronic Resource
    Electronic Resource
    Accuracy and stability of multistage multistep formulas (1986)
    Springer
    Numerische Mathematik 48 (1986), S. 33-83 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65L05, 65L07 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We continue our research concerning relations between accuracy and stability of time discretizations. Here we concentrate, rather than on the size and shape of the stability region, on the obtainable accuracy when the number of parameters in the formulas is large.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389442
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  • 2
    Electronic Resource
    Electronic Resource
    A 0-Stability and stiff stability of Brown's multistep multiderivative methods (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 167-181 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Brown [1] introducedk-step methods usingl derivatives. Necessary and sufficient conditions forA 0-stability and stiff stability of these methods are given. These conditions are used to investigate for whichk andl the methods areA 0-stable. It is seen that for allk andl withk≦1.5 (l+1) the methods areA 0-stable and stiffly stable. This result is conservative and can be improved forl sufficiently large. For smallk andl A 0-stability has been determined numerically by implementing the necessary and sufficient condition.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01404873
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  • 3
    Electronic Resource
    Electronic Resource
    Convergence analysis of discretization methods for nonlinear first kind Volterra integral equations (1986)
    Springer
    Numerische Mathematik 49 (1986), S. 67-80 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 R 20 ; CR: G1.9
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary An existence and uniqueness result is given for nonlinear Volterra integral equations of the first kind. This permits, by means of analogous discrete manipulations, a general convergence analysis for a wide class of discretization methods for nonlinear first kind Volterra integral equations to be presented. A concept of optimal consistency allows twosided error bounds to be derived.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389430
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Integration of iterated integrals by multistep methods (1973)
    Springer
    Numerische Mathematik 21 (1973), S. 303-316 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Linear multistep methods with looking ahead points are discussed for the class of initial value problems:d r y/d x r =f(x). Upper bounds for the error order implied by the condition of stability are given for methods involving (i)f(x) only and (ii) bothf(x) andf′(x). The methods discussed are more general than those treated by Dahlquist. It is seen that in case (i) the bounds depend uponr while in case (ii) these are independent ofr.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01436385
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  • 5
    Electronic Resource
    Electronic Resource
    On the stability properties of Brown's multistep multiderivative methods (1978)
    Springer
    Numerische Mathematik 30 (1978), S. 25-38 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Brown introducedk-step methods usingl derivatives. We investigate for whichk andl the methods are stable or unstable. It is seen that to anyl the method becomes unstable fork large enough. All methods withk≦2(l+1) are stable. Fork=1,2,..., 18 there exists aλ k such that the methods are stable for anyl ≧λ k and unstable for anyl 〈λ k . Theλ k are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01403904
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    Stability of explicit time discretizations for solving initial value problems (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 61-91 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Stability regions of explicit “linear” time discretization methods for solving initial value problems are treated. If an integration method needsm function evaluations per time step, then we scale the stability region by dividing bym. We show that the scaled stability region of a method, satisfying some reasonable conditions, cannot be properly contained in the scaled stability region of another method. Bounds for the size of the stability regions for three different purposes are then given: for “general” nonlinear ordinary differential systems, for systems obtained from parabolic problems and for systems obtained from hyperbolic problems. We also show how these bounds can be approached by high order methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396187
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Counterexamples to a stability barrier (1987)
    Springer
    Numerische Mathematik 52 (1987), S. 301-316 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65M10 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider multistep difference schemes for the linear, constant coefficient advection equationu t=cux. In the last section of Strang [5], a barrier for the order of stable schemes has been given which was independent of the number of time levels. Here we give two types of counterexamples to this barrier. The first type consists of formulas of highest possible order to a given stencil, which are stable for small positive Courant numbers. Further a formula is given where one does not insist on having the highest possible order for the stencil and uses the gained freedom to ensure stability for small positive and negative Courant numbers. In addition, an explicit formula is derived for the schemes of highest possible order when stability is disregarded.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01398881
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    Stability and accuracy of time discretizations for initial value problems (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 245-296 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems. Here the treatment is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general ‘method-free’ statements are again obtained. More specialized results are additionally given for linear multistep methods and for the Taylor series method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400542
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    Paper (German National Licenses)
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  • 9
    Electronic Resource
    Electronic Resource
    Stability of a family of multi-time-level difference schemes for the advection equation (1991)
    Springer
    Numerische Mathematik 60 (1991), S. 77-95 
    Details
    ISSN: 0945-3245
    Keywords: 65M10 ; 65N05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For the linear advection equation we consider explicit multi-time-level schemes of highest order which are one step in space direction only. If a stencil involvesk time steps we show that it is stable in theL 2-sense for Courant numbers in the interval (0, 1/k). Since the order is 2k−1 one can use these schemes for high order discretization of the boundary conditions in hyperbolic initial value problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385715
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    Paper (German National Licenses)
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  • 10
    Electronic Resource
    Electronic Resource
    Buchbesprechungen (1988)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 39 (1988), S. 129-134 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00945728
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    Paper (German National Licenses)
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